Math Problem Statement

Determine the key features of the graph of the function f(x) = ((x - 1)(x + 1))/(x + 2), including zeros, vertical asymptotes, and horizontal asymptotes.

Solution

To analyze the function:

f(x)=(x−1)(x+1)x+2,f(x) = \frac{(x - 1)(x + 1)}{x + 2},

we will determine key features of the graph, including the zeros, vertical asymptotes, and horizontal asymptotes.

Step 1: Zeros of f(x)f(x)

Zeros occur when the numerator equals zero, while the denominator is non-zero. For the numerator:

(x−1)(x+1)=0(x - 1)(x + 1) = 0

The solutions are: [ x = 1 \

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Math Problem Analysis

Mathematical Concepts

Algebra
Rational Functions
Graphing
Asymptotes
Zeros of Functions

Formulas

f(x) = (x - 1)(x + 1) / (x + 2)
Conditions for zeros: Numerator = 0 while Denominator ≠ 0
Conditions for vertical asymptotes: Denominator = 0

Theorems

Fundamental Theorem of Algebra
Behavior of Rational Functions at Asymptotes

Suitable Grade Level

Grades 10-12

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